Smile Model Research Articles

Philosophy of finance. Definitive smile model: Part I

Philosophy of finance. Definitive smile model: Part I

Candidate Market Models and the Calibrated CIR++ Stochastic Intensity Model for Credit Default Swap Options and Callable Floaters

We consider the standard Credit Default Swap (CDS) payoff and some alternative approximated versions, stemming from different conventions on the premium and protection legs. We consider standard running CDS (RCDS), upfront CDS and postponed-payments running CDS (PRCDS). Each different definition implies a different definition of forward CDS rate, which we consider with some detail. We introduce defaultable floating rate notes (FRN)'s. We point out which kind of CDS payoff produces a forward CDS rate that is equal to the fair spread in the considered defaultable FRN. An approximated equivalence between CDS's and defaultable FRN's is established, which allows to view CDS options as structurally similar to the optional component in defaultable callable notes. We briefly investigate the possibility to express forward CDS rates in terms of some basic rates and discuss a possible analogy with the LIBOR and swap default free models. Finally, we discuss the change of numeraire approach for deriving a Black-like formula for CDS options or, equivalently, defaultable callable FRN's. We also introduce an analytical formula for CDS option prices under the CDS-calibrated SSRD stochastic-intensity model, and discuss the impact of the different CIR++ dynamics parameters on the related CDS options implied volatilities. Hints on possible methods for smile modeling of CDS options are given for possible future developments of the CDS option market.

The performance of alternative valuation models in the OTC currency options market

We compare option valuation models based on regime-switching, GARCH, and jump-diffusion processes to a standard “smile” model, in which Black and Scholes (1973) implied volatilities are allowed to vary across strike prices. The regime-switching, GARCH, and jump-diffusion models provide significant improvement over a fixed smile model in fitting GBP and JPY option prices both in-sample and out-of-sample. The jump-diffusion model achieves the tightest fit. A time-varying smile model, however, provides hedging performance that is comparable to the other models for the GBP options. This result suggests that standard option valuation techniques may provide a reasonable basis for trading and hedging strategies.

A one-factor volatility smile model with closed-form solutions for European options

The common practice of using different volatilities for options of different strikes in the Black-Scholes (1973) model imposes inconsistent assumptions on underlying securities. The phenomenon is referred to as the volatility smile. This paper addresses this problem by replacing the Brownian motion or, alternatively, the Geometric Brownian motion in the Black-Scholes model with a two-piece quadratic or linear function of the Brownian motion. By selecting appropriate parameters of this function we obtain a wide range of shapes of implied volatility curves with respect to option strikes. The model has closed-form solutions for European options, which enables fast calibration of the model to market option prices. The model can also be efficiently implemented in discrete time for pricing complex options. G1